L10a44

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L10a43.gif

L10a43

L10a45.gif

L10a45

Contents

L10a44.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10a44 at Knotilus!


Link Presentations

[edit Notes on L10a44's Link Presentations]

Planar diagram presentation X6172 X14,3,15,4 X16,8,17,7 X18,10,19,9 X20,12,5,11 X8,18,9,17 X10,20,11,19 X12,16,13,15 X2536 X4,13,1,14
Gauss code {1, -9, 2, -10}, {9, -1, 3, -6, 4, -7, 5, -8, 10, -2, 8, -3, 6, -4, 7, -5}
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10a44 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(2)^5-2 t(2)^5-2 t(1) t(2)^4+2 t(2)^4+2 t(1) t(2)^3-2 t(2)^3-2 t(1) t(2)^2+2 t(2)^2+2 t(1) t(2)-2 t(2)-2 t(1)+1}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -q^{15/2}+3 q^{13/2}-4 q^{11/2}+6 q^{9/2}-7 q^{7/2}+7 q^{5/2}-6 q^{3/2}+4 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{1}{q^{5/2}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z^7 a^{-3} -2 z^5 a^{-1} +5 z^5 a^{-3} -z^5 a^{-5} +a z^3-9 z^3 a^{-1} +8 z^3 a^{-3} -3 z^3 a^{-5} +4 a z-11 z a^{-1} +6 z a^{-3} -z a^{-5} +3 a z^{-1} -5 a^{-1} z^{-1} +2 a^{-3} z^{-1} (db)
Kauffman polynomial z^3 a^{-9} +3 z^4 a^{-8} -2 z^2 a^{-8} +4 z^5 a^{-7} -3 z^3 a^{-7} +4 z^6 a^{-6} -3 z^4 a^{-6} -2 z^2 a^{-6} + a^{-6} +4 z^7 a^{-5} -7 z^5 a^{-5} +3 z^3 a^{-5} -z a^{-5} +3 z^8 a^{-4} -7 z^6 a^{-4} +4 z^4 a^{-4} -z^2 a^{-4} +z^9 a^{-3} +z^7 a^{-3} -11 z^5 a^{-3} +11 z^3 a^{-3} -5 z a^{-3} +2 a^{-3} z^{-1} +4 z^8 a^{-2} -14 z^6 a^{-2} +8 z^4 a^{-2} +8 z^2 a^{-2} -5 a^{-2} +z^9 a^{-1} +a z^7-2 z^7 a^{-1} -6 a z^5-6 z^5 a^{-1} +12 a z^3+16 z^3 a^{-1} -10 a z-14 z a^{-1} +3 a z^{-1} +5 a^{-1} z^{-1} +z^8-3 z^6-2 z^4+9 z^2-5 (db)

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r).   
\ r
  \  
j \
-4-3-2-10123456χ
16          11
14         2 -2
12        21 1
10       42  -2
8      32   1
6     44    0
4    23     -1
2   35      2
0  11       0
-2  3        3
-411         0
-61          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=2 i=4
r=-4 {\mathbb Z} {\mathbb Z}
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\mathbb Z}_2 {\mathbb Z}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

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See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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L10a45